An approach to sub-Gaussian heat kernel estimates via analysis on metric spaces

In this work, we establish a new characterization of sub-Gaussian heat kernel estimates for strongly local regular Dirichlet forms on metric measure spaces. Our formulation is based on the newly introduced cutoff energy condition, which offers a simpler and more transparent alternative for earlier technical energy inequalities, in particular the cutoff Sobolev inequality. The main idea of our approach is to reinterpret the cutoff Sobolev inequality as a Poincaré type inequality, and analyze it using Hajłasz–Koskela techniques from analysis on metric spaces. Applications of the new characterization are also discussed.